Use the Product Rule of Exponents to simplify 3^5 times 3^7. show your work.

Answer 1

The Product Rule of Exponents states that when you multiply two powers with the same base, you can add their exponents. The rule can be expressed as:

\[ a^m \times a^n = a^{m+n} \]

In your case, you have \(3^5\) and \(3^7\), which both have the same base of 3.

Using the Product Rule:

\[ 3^5 \times 3^7 = 3^{5+7} \]

Now, add the exponents:

\[ 5 + 7 = 12 \]

So, we can simplify:

\[ 3^5 \times 3^7 = 3^{12} \]

Therefore, the simplified expression is:

\[ 3^{12} \]

Answer 2

Find the numerical equivalent of 5^10 times 5^-6. The answer should not contain any exponent expression or decimals. Show your work.

Answer 3

To simplify \(5^{10} \times 5^{-6}\), we can use the Product Rule of Exponents, which states:

\[ a^m \times a^n = a^{m+n} \]

In this case:

\[ 5^{10} \times 5^{-6} = 5^{10 + (-6)} = 5^{10 - 6} = 5^{4} \]

Now, we need to find the numerical equivalent of \(5^4\).

Calculating \(5^4\):

\[ 5^4 = 5 \times 5 \times 5 \times 5 \]

Calculating step by step:

1. \(5 \times 5 = 25\)
2. \(25 \times 5 = 125\)
3. \(125 \times 5 = 625\)

So,

\[ 5^4 = 625 \]

Thus, the numerical equivalent of \(5^{10} \times 5^{-6}\) is:

\[ \boxed{625} \]